- Hacettepe Journal of Mathematics and Statistics
- Vol: 48 Issue: 4
- Generalized Lucas numbers of the form $11x^{2}\mp 1$
Generalized Lucas numbers of the form $11x^{2}\mp 1$
Authors : Refik Keskin, Ümmügülsüm Öğüt
Pages : 1035-1045
View : 12 | Download : 6
Publication Date : 2019-08-08
Article Type : Research
Abstract :Let $P\geq3$ be an integer and $(V_{n})$ denote generalized Lucas sequence defined by $V_{0}=2,V_{1}=P,$ and $V_{n+1}=PV_{n}-V_{n-1}$ for $n\geq1.$ In this study, we solve the equation $V_{n}=11x^{2}\mp1.$ We show that the equation $V_{n}=11x^{2}+1$ has a solution only when $n=1$ and $P\equiv 1({mod}11)$. Moreover, we show that if the equation $V_{n}=11x^{2}-1$ has a solution, then $P\equiv2({mod}8)$ and $P\equiv-1({mod}11).$Keywords : generalized Lucas numbers, congruences, diophantine equation